Hey, everyone. Welcome back. So we've multiplied and divided radicals, and one of the things that we should know about radicals is that they can never be left in the bottom of a fraction. This is one of those weird bad things that you just can't do in math. Now you might be thinking we've already seen radicals in the bottoms of fractions like √2 of √8. But in that case, it was fine because usually those fractions reduced to perfect squares like 1 over 4. And then if it was a perfect square, the radical just goes away, and you're left with a rational number. What I'm gonna show you in this video is that sometimes that doesn't happen. Sometimes you might have an expression like 1 over √3, and you can't simplify that to a perfect square. So to solve these types of problems, we're gonna have to do another thing. We're gonna have to do something called rationalizing the denominator. I'm gonna show you what that process is. It's actually really straightforward, so let's just go ahead and get to it. So again, if we have something like √2 of √8, it's simplified to a perfect square, and that was perfectly fine. So radicals can simplify to perfect squares, and we don't have to do anything else because you're just left with something like 1 half. But if you can't simplify this radical over here to a perfect square, then we're gonna have to make it 1. And the way we make it 1 is by doing this thing called rationalizing the denominator. It's actually really straightforward. Basically, we're gonna take this expression over here, and we're gonna multiply it by something to get rid of that radical on the bottom. And so what you're gonna do is you're gonna multiply the top and the bottom, the numerator and the denominator, by something, and, usually, that something that you multiply by is just whatever is on the bottom radical. So, in other words, we're gonna take this expression over here, and and I'm just gonna multiply it by √3, but I have to do it on the top and the bottom. You always have to make sure to do it on the top and the bottom because then you're basically just multiplying this expression by 1, and you're not changing the value of it. So whatever you do at the bottom, you have to do on the top. And the reason this works is because let's just work it out. What is √3 × √3? Basically, once we've done this, we've now turned the bottom into a perfect square. It's the square root of 9, which we know is actually just 3. So in other words, we've multiplied it by itself to sort of get rid of the radical, and now it's just a rational number on the bottom. Alright? So what happens to the top? Well, again, we just multiply straight across, and then we ended up with √3 over √9, which is just √3 over 3. So, look at the difference between where we started and ended. Here, we had 1 over √3, we had a radical on the bottom. And here, when we're done, we actually have 3 on the bottom and that's perfectly fine. We have a radical on top, but we can have radicals on the top, and that's perfectly fine. So what I want you to do is I actually want you to plug in, if you have a calculator handy, 1 divided by √3. When you plug this in, what you should get out of the calculator is 0.57. And now if you actually do √3 over 3, you're gonna get the exact same numbers, 0.57. So the whole thing here is that these two expressions are exactly equivalent. They mean the exact same thing. It's just that in one case, we've gotten rid of the radical and the bottom. So this is what rationalizing the denominator means. Thanks for watching, and let's move on to the next one.